We study the regularity of the Euclidean distance function from a given point-wise target of a n-dimensional vector space in the presence of a compact obstacle bounded by a smooth hypersurface. It is known that such a function is semiconcave with fractional modulus one half. We provide a geometrical explanation of the exponent one half. Furthermore, under a natural (weak) assumption on the position of the point-wise target relatively to the obstacle, we show that there exists a point in the boundary of the obstacle so that no better regularity result holds near such a point. As a consequence of this result, we show that the Euclidean metric cannot be extended to a tubular neighborhood of the obstacle, as a Riemannian metric, keeping the property that the associated distances coincide outside the obstacle.

Albano P. (2023). On the regularity of the distance near the boundary of an obstacle. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 518(1), 1-12 [10.1016/j.jmaa.2022.126680].

### On the regularity of the distance near the boundary of an obstacle

#### Abstract

We study the regularity of the Euclidean distance function from a given point-wise target of a n-dimensional vector space in the presence of a compact obstacle bounded by a smooth hypersurface. It is known that such a function is semiconcave with fractional modulus one half. We provide a geometrical explanation of the exponent one half. Furthermore, under a natural (weak) assumption on the position of the point-wise target relatively to the obstacle, we show that there exists a point in the boundary of the obstacle so that no better regularity result holds near such a point. As a consequence of this result, we show that the Euclidean metric cannot be extended to a tubular neighborhood of the obstacle, as a Riemannian metric, keeping the property that the associated distances coincide outside the obstacle.
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2023
Albano P. (2023). On the regularity of the distance near the boundary of an obstacle. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 518(1), 1-12 [10.1016/j.jmaa.2022.126680].
Albano P.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/901223`